Question

Discuss the continuity of $f\left(x\right)=\sin \left|x\right|.$

Answer 1

Given,

$f\left(x\right)=\sin \left|x\right|$

Let $a$ be any real number.

Now, LHL at $x=a$,

$\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(a-h)=\underset{h\to 0}{lim}\sin |a-h|=\sin \left|a\right|$

RHL at $x=a$,

$\underset{x\to a^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(a+h)=\underset{h\to 0}{lim}\sin |a+h|=\sin \left|a\right|$

Also,

$f\left(a\right)=\sin \left|a\right|$

$\Rightarrow \underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(a\right)$

Thus $f\left(x\right)$ is continuous at $x=a$.

Since $a$ is an arbitrary real number, hence $f\left(x\right)$ is continuous everywhere.